To attend​ school, 

RalphRalph 

deposits 

​$720720 

at the end of every 

quarterquarter 

for 

fivefive 

and​ one-half years. What is the accumulated value of the deposits if interest is 

33​% 

compounded 

monthlymonthly​?

Question content area bottom

Part 1

The accumulated value is 

​$enter your response here. 

​(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as​ needed.)

To calculate the accumulated value of the deposits, we use the future value of an annuity formula. However, because the deposits are made quarterly and the interest is compounded monthly, we need to account for the different compounding periods.

### Step 1: Identify the values
- $ P = 720 $ (amount of each deposit)
- Annual interest rate $ = 3\% = 0.03 $
- Compounding frequency = 12 times per year (monthly compounding)
- Number of years $ = 5.5 $ (five and one-half years)

### Step 2: Calculate the interest rate per period and the total number of periods
- Since interest is compounded monthly, the interest rate per period (monthly) is:
  \[
  r = \frac{0.03}{12} = 0.0025
  \]
- The total number of periods (in months) is:
  \[
  n = 5.5 \times 12 = 66 \text{ months}
  \]

### Step 3: Adjust for the quarterly deposits
Since deposits are made quarterly (every 3 months), each deposit actually represents a series of deposits spaced 3 months apart. To handle this, we'll adjust the annuity formula:

The formula to calculate the future value of a series of deposits made every $ k $ periods (in this case, every 3 months) with monthly compounding is:
\[
FV = P \times \left(\frac{(1 + r)^n - 1}{(1 + r)^k - 1}\right)
\]
Where $ k = 3 $ for quarterly deposits.

### Step 4: Plug in the values and calculate
Let's calculate this step by step:

1. $ (1 + 0.0025)^{66} = 1.171659 $ (rounded to six decimal places)
2. $ (1.171659 - 1) = 0.171659 $
3. $ (1 + 0.0025)^{3} = 1.007528 $ (rounded to six decimal places)
4. $ (1.007528 - 1) = 0.007528 $
5. $ \frac{0.171659}{0.007528} = 22.810505 $
6. Finally, $ 720 \times 22.810505 = 16423.564 $

### Step 5: Round to the nearest cent
The final accumulated value is approximately **\$16,423.56**.

Would you like any further details or have any questions?

---

Here are some related questions:
1. How would the accumulated value change if the interest rate was increased to 5%?
2. What if Ralph made monthly deposits instead of quarterly?
3. How much would the future value be if Ralph only deposited for 4 years?
4. How does the compounding frequency (monthly vs. quarterly) affect the accumulated value?
5. What would be the future value if the deposit amount was $1,000 instead of $720?
6. How much interest will Ralph have earned by the end of 5.5 years?
7. What if the interest was compounded quarterly instead of monthly?
8. How would the result differ if the interest rate was reduced to 2%?

**Tip:** When working with different compounding periods and deposit intervals, be careful to adjust the formula to account for these differences.

Learn how to calculate the accumulated value of quarterly deposits over 5.5 years with a 33% annual interest rate compounded monthly. Detailed step-by-step solution using the future value of an annuity formula. Suitable for high school students studying compound interest and financial mathematics.

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